cpmadumatpolylem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cpmadumatpolylem2		Structured version Visualization version GIF version

Theorem cpmadumatpolylem2 22102

Description: Lemma 2 for cpmadumatpoly 22103. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

**Hypotheses**
Ref	Expression
cpmadumatpoly.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadumatpoly.b	⊢ 𝐵 = (Base‘𝐴)
cpmadumatpoly.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmadumatpoly.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadumatpoly.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadumatpoly.r	⊢ × = (._r‘𝑌)
cpmadumatpoly.m0	⊢ − = (-_g‘𝑌)
cpmadumatpoly.0	⊢ 0 = (0_g‘𝑌)
cpmadumatpoly.g	⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
cpmadumatpoly.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmadumatpoly.m1	⊢ · = ( ·_𝑠 ‘𝑌)
cpmadumatpoly.1	⊢ 1 = (1_r‘𝑌)
cpmadumatpoly.z	⊢ 𝑍 = (var₁‘𝑅)
cpmadumatpoly.d	⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))
cpmadumatpoly.j	⊢ 𝐽 = (𝑁 maAdju 𝑃)
cpmadumatpoly.w	⊢ 𝑊 = (Base‘𝑌)
cpmadumatpoly.q	⊢ 𝑄 = (Poly₁‘𝐴)
cpmadumatpoly.x	⊢ 𝑋 = (var₁‘𝐴)
cpmadumatpoly.m2	⊢ ∗ = ( ·_𝑠 ‘𝑄)
cpmadumatpoly.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
cpmadumatpoly.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)

**Assertion**
Ref	Expression
cpmadumatpolylem2	⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑆,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠 Allowed substitution hints: 𝐴(𝑛,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝐷(𝑛,𝑠,𝑏) 𝑃(𝑛,𝑠,𝑏) 𝑄(𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑆(𝑠,𝑏) 𝑇(𝑛,𝑠,𝑏) · (𝑛,𝑠,𝑏) × (𝑛,𝑠,𝑏) 𝑈(𝑛,𝑠,𝑏) 1 (𝑛,𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) ∗ (𝑛,𝑠,𝑏) 𝐽(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑛,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑊(𝑛,𝑠,𝑏) 𝑋(𝑛,𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑛,𝑠,𝑏) 𝑍(𝑛,𝑠,𝑏)

**Proof of Theorem cpmadumatpolylem2**
Step	Hyp	Ref	Expression
1		fvexd 6824	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝐴) ∈ V)
2		crngring 19862	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim2i 617	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	3adant3 1131	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5	4	ad2antrr 723	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6		cpmadumatpoly.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
7		cpmadumatpoly.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
8		cpmadumatpoly.y	. . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	6, 7, 8	0elcpmat 21942	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝑌) ∈ 𝑆)
10	5, 9	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝑌) ∈ 𝑆)
11		cpmadumatpoly.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12		cpmadumatpoly.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
13		cpmadumatpoly.r	. . . . 5 ⊢ × = (._r‘𝑌)
14		cpmadumatpoly.m0	. . . . 5 ⊢ − = (-_g‘𝑌)
15		cpmadumatpoly.0	. . . . 5 ⊢ 0 = (0_g‘𝑌)
16		cpmadumatpoly.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
17		cpmadumatpoly.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
18	11, 12, 7, 8, 13, 14, 15, 16, 17, 6	chfacfisfcpmat 22075	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
19	2, 18	syl3anl2 1412	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
20	19	anassrs 468	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺:ℕ₀⟶𝑆)
21		cpmadumatpoly.u	. . . 4 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22	11, 12, 6, 21	cpm2mf 21972	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:𝑆⟶𝐵)
23	5, 22	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑈:𝑆⟶𝐵)
24		ssidd 3953	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ⊆ 𝑆)
25		nn0ex 12309	. . 3 ⊢ ℕ₀ ∈ V
26	25	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → ℕ₀ ∈ V)
27	6	ovexi 7347	. . 3 ⊢ 𝑆 ∈ V
28	27	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ∈ V)
29	11, 12, 7, 8, 13, 14, 15, 16, 17	chfacffsupp 22076	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))
30	29	anassrs 468	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺 finSupp (0_g‘𝑌))
31		eqid 2737	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
32		eqid 2737	. . . . . 6 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
33	11, 21, 7, 8, 31, 32	m2cpminv0 21981	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
34	2, 33	sylan2 593	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
35	34	3adant3 1131	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
36	35	ad2antrr 723	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
37	1, 10, 20, 23, 24, 26, 28, 30, 36	fsuppcor 9231	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ifcif 4469 class class class wbr 5085 ↦ cmpt 5168 ∘ ccom 5609 ⟶wf 6459 ‘cfv 6463 (class class class)co 7313 ↑_m cmap 8661 Fincfn 8779 finSupp cfsupp 9196 0cc0 10941 1c1 10942 + caddc 10944 < clt 11079 − cmin 11275 ℕcn 12043 ℕ₀cn0 12303 ...cfz 13309 Basecbs 16979 ._rcmulr 17030 ·_𝑠 cvsca 17033 0_gc0g 17217 -_gcsg 18646 ._gcmg 18767 mulGrpcmgp 19787 1_rcur 19804 Ringcrg 19850 CRingccrg 19851 var₁cv1 21418 Poly₁cpl1 21419 Mat cmat 21625 maAdju cmadu 21852 ConstPolyMat ccpmat 21923 matToPolyMat cmat2pmat 21924 cPolyMatToMat ccpmat2mat 21925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4849 df-int 4891 df-iun 4937 df-iin 4938 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-se 5561 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-isom 6472 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-ofr 7572 df-om 7756 df-1st 7874 df-2nd 7875 df-supp 8023 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-er 8544 df-map 8663 df-pm 8664 df-ixp 8732 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-fsupp 9197 df-sup 9269 df-oi 9337 df-card 9765 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-7 12111 df-8 12112 df-9 12113 df-n0 12304 df-z 12390 df-dec 12508 df-uz 12653 df-fz 13310 df-fzo 13453 df-seq 13792 df-hash 14115 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-sca 17045 df-vsca 17046 df-ip 17047 df-tset 17048 df-ple 17049 df-ds 17051 df-hom 17053 df-cco 17054 df-0g 17219 df-gsum 17220 df-prds 17225 df-pws 17227 df-mre 17362 df-mrc 17363 df-acs 17365 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-mhm 18497 df-submnd 18498 df-grp 18647 df-minusg 18648 df-sbg 18649 df-mulg 18768 df-subg 18819 df-ghm 18899 df-cntz 18990 df-cmn 19455 df-abl 19456 df-mgp 19788 df-ur 19805 df-srg 19809 df-ring 19852 df-cring 19853 df-subrg 20093 df-lmod 20196 df-lss 20265 df-sra 20505 df-rgmod 20506 df-dsmm 21010 df-frlm 21025 df-ascl 21133 df-psr 21183 df-mvr 21184 df-mpl 21185 df-opsr 21187 df-psr1 21422 df-vr1 21423 df-ply1 21424 df-coe1 21425 df-mamu 21604 df-mat 21626 df-cpmat 21926 df-mat2pmat 21927 df-cpmat2mat 21928

This theorem is referenced by: cpmadumatpoly 22103 chcoeffeqlem 22105

W3C validator